User:Prof McCarthy/Kinematics equations

Kinematics equations refers to the constraint equations of a mechanical system such as a robot manipulator that define the input movement at one or more joints that control the overall configuration of the device, in order to achieve a task pro position an end-effector. Kinematics equations are used to analyze and design articulated systems ranging from four-bar linkages to serial and parallel robots.

Kinematics equations are constraint equations that characterize the geometric configuration of an articulated mechanical system. Therefore, these equations are based on the assumptions that the links are rigid and the joints provide pure rotation or translation. Constraint equations of this type are known as holonomic constraints in the study of the dynamics of multi-body systems.

Loop equations
The kinematics equations for a mechanical system are formulated as a sequence of rigid transformations along links and around joints in an articulated system. The principle that the sequence of transformations around a loop must return to the identity provides what are known as the loop equations. An independent set of kinematics equations is assembled from the various sets of loop equations that characterize a mechanical device.

Transformations
n 1955, Jacques Denavit and Richard Hartenberg introduced a convention for the definition of the joint matrices [Z] and link matrices [X] to standardize the coordinate frames for spatial linkages. This convention positions the joint frame so that it consists of a screw displacement along the Z-axis
 * $$ [Z_i] = \begin{bmatrix}

\cos\theta_i & -\sin\theta_i & 0 & 0 \\ \sin\theta_i & \cos\theta_i & 0  & 0 \\ 0 & 0 & 1 & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix}, $$

and it positions the link frame so it consists of a screw displacement along the X-axis,
 * $$ [X_i]=\begin{bmatrix}

1 & 0 & 0 & a_{i,i+1} \\ 0 & \cos\alpha_{i,i+1} & -\sin\alpha_{i,i+1}& 0 \\ 0 & \sin\alpha_{i,i+1} & \cos\alpha_{i,i+1} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}. $$

The kinematics equations are obtained using a rigid transformation [Z] to characterize the relative movement allowed at each joint and separate rigid transformation [X] to define the dimensions of each link.

The result is a sequence of rigid transformations alternating joint and link transformations from the base of the chain around a loop back to the base to obtain the loop equation,
 * $$[Z_1][X_1][Z_2][X_2]\ldots[X_{n-1}][Z_n]=[I].\!$$

The series of transformations equate to the identify matrix because they return to the beginning of the loop.

Serial chains
The kinematics equations for a serial chain robot are obtained by formulating the loop equations in terms of a transformation [T] from the base to the end-effector, which is equated to the series of transformations along the robot. The result is,
 * $$[T] = [Z_1][X_1][Z_2][X_2]\ldots[X_{n-1}][Z_n],\!$$

These equations are called the kinematics equations of the serial chain.